Greenfüggvények

Greenfüggvények, or Green's functions, are a powerful mathematical tool used in solving linear differential equations with given boundary conditions. They represent the response of a system to a point source or impulse. In essence, a Green's function is a solution to a linear differential equation where the source term is a Dirac delta function, which is zero everywhere except at a single point, where it is infinitely large in such a way that its integral is one.

The fundamental idea behind using Green's functions is that any general solution to a linear differential equation can be constructed by integrating the Green's function multiplied by the actual source term. This is because the Dirac delta function is a basis for all functions, meaning any function can be represented as a sum of scaled and shifted Dirac delta functions. Therefore, if one knows the response to a single delta function, the response to any arbitrary source can be found by summing up the responses to these individual delta functions, weighted by the strength of the source at each point.

Green's functions are widely applied in various fields of physics and engineering, including electromagnetism, quantum mechanics, wave propagation, and heat transfer. They simplify the process of solving inhomogeneous differential equations, particularly when dealing with complex geometries or boundary conditions. The systematic approach provided by Green's functions allows for the derivation of analytical solutions or facilitates numerical approximations for problems that would otherwise be intractable.